In the form of (), the discrete vorticity evolution equation does not contain any transport stabilisation measures, and we may therefore expect it to be vulnerable to grid-scale oscillations.
In the context of a shallow-water model this can correspond to a lack of dissipation of enstrophy, which naturally cascades to fine scales but gets trapped at the grid scale without a mechanism to dissipate it [1]}.
Since the vorticity is discretised as a \(\mathrm {CG}_k\)  field, this can be remedied by using a stabilisation based on the SUPG method.
